I might not understand exactly what you are saying. Are you saying that the problem is easy when you have a function that gives you the coordinates of an arbitrary node? Isn’t that exactly the embedding function? So are you not therefore assuming that you have an embedding function?
I agree that once you have such a function the problem is easy, but I am confused about how you are getting that function in the first place. If you are not given it, then I don’t think it is super easy to get.
In the OP I was assuming that I have that function, but I was saying that this is not a valid assumption in general. You can imagine you are just given a set of vertices and edges. Now you want to compute the embedding such that you can do the vector planning described in the article.
I agree that you probably can do better than 10100 though. I don’t understand how your proposal helps though.
Do you want me to spoil it for you, do you want me to drop a hint, or do you want to puzzle it out yourself? It’s a beautiful little puzzle and very satisfying to solve. Also note that the solution I found only works if you are given a graph with the structure above (i.e. every node is part of the lattice, and the lattice is fairly small in each dimension, and the lattice has edges rather than wrapping around).
I might not understand exactly what you are saying. Are you saying that the problem is easy when you have a function that gives you the coordinates of an arbitrary node? Isn’t that exactly the embedding function? So are you not therefore assuming that you have an embedding function?
I agree that once you have such a function the problem is easy, but I am confused about how you are getting that function in the first place. If you are not given it, then I don’t think it is super easy to get.
In the OP I was assuming that I have that function, but I was saying that this is not a valid assumption in general. You can imagine you are just given a set of vertices and edges. Now you want to compute the embedding such that you can do the vector planning described in the article.
I agree that you probably can do better than 10100 though. I don’t understand how your proposal helps though.
Do you want me to spoil it for you, do you want me to drop a hint, or do you want to puzzle it out yourself? It’s a beautiful little puzzle and very satisfying to solve. Also note that the solution I found only works if you are given a graph with the structure above (i.e. every node is part of the lattice, and the lattice is fairly small in each dimension, and the lattice has edges rather than wrapping around).